1.1.1—Polynomial roots

For ax²+bx+c=0 with roots α and β, α+β=−b/a and αβ=c/a. These relations are often faster and safer than applying the quadratic formula twice.

For higher-degree polynomials, factorisation and the factor theorem identify roots; repeated roots occur when a factor is squared or the polynomial and its derivative share a root. Keep the equation in standard form before reading coefficients.

For 2x²−5x+3=0, the roots have sum 5/2 and product 3/2, so their average is 5/4 without finding either root.

The coefficient relationships depend on signs and the leading coefficient; they are not simply “sum=b, product=c”.